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An Investigation into Model Order Reduction through
Balancing Methods and their Error Norm
Deepak Gupta
1, Awadhesh Kumar
21,2Electrical Engineering Department, MMMUT, Gorakhpur, India
Abstract-- In simulation (control) one seeks to predict (modify) the system behavior; however, computation of the complete model is often not practicable, necessitating simplification of it. Due to finite computational, correctness, and storage capability, system approximation the development of simplified models that capture the soul features of the original dynamical systems evolve. Simplified models are used in place of original complex models and result in control with reduced the complexity problem.
A central concept in system theory with application to reduce the order of model through balanced representation of a system roughly speaking, the states in such a representation are such that the scale of reachability and the scale of observability of each state are the same. So we have presented the various balancing methods used in model order reduction with its corresponding error norms.
I. INTRODUCTION
Suppose the LTI (dynamical) systems in time-domain form (state space form)
0 0 0
0 0
0 0
0 0
1
0 0 0 0
( )
( )
( )
( ) :
( ) :
( )
( )
( )
:
(
)
(1)
o
x t
A x t
B u t
A
B
G s
G s
C
D
y t
C x t
D u t
TF
C sI
A
B
D
where A,B,C,D are matrices of different order i.e.
,
,
,
.
n n n m p n p m
A
B
C
D
In many of the practical case the order of the system i.e. n are very large while the system may be single input single output (SISO) or multiple input multiple output (MIMO) must satisfies m,p
n
. For all such type of system there are various problem occurs such as computational problem because of larger memory demand, time limitation and other conditions. One of the solutions of above problem is solved by model reduction.The problem related to model order reduction can be described as follows i.e. the reduced system of order and the matrices of order
respectively, with satisfying the certain property
i.e.
1 The approximation error is small, and there contain a global error bound.
2 System properties, like stability, passivity, are converged.
3 The procedure is computationally competent.
One model reduction scheme that is well defined in theory and most widely used is termed as balanced model reduction first introduced by Mullis and Roberts (1976) and later described by Moore (1981) in the systems and control literature. To apply balanced reduction, first the system is transformed to balance basis where the states which are difficult to reach and at the same time difficult to observe. This is obtained by together the controllability and the observability gramians, which are solutions to the controllability and the observability Lyapunov equations. Then, by truncating the states which have this property we obtained the reduced model.
II. LYAPUNOV BALANCING METHOD
When applied to stable systems, Lyapunov balanced reduction preserves stability [13] and provides a bound on the approximation error [4], i.e. satisfies (i) and (ii) above. For small and medium scale problems, Lyapunov balancing can be implemented practically. However, for large scale settings, it requires dense matrix factorizations and results in a simulating complexity of O ( ) and a memory capability of O ( ); hence does not satisfy (iii) above. In this case, balanced reduction is a recent research area through which a numerically efficient way is to obtain in the field of model reduction.
Let
( ) * + ( ) ( )
Be the original model which is the to-be-reduced model as defined in (1). The given two continuous time Lyapunov equations is connected to above system i.e.
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Let G(s) is asymptotically stable and minimal; the given two Lyapunov equations contain a property that has unique symmetric positive definite solutions ( ) termed as controllability and observability gramians, respectively. The square roots of the eigenvalues of the product controllability and observability gramians (PQ) are the so-called Hankel singular value ( ( )) of the G(s):
( ( )) √ ( )
It is found that ( ( )) are basis independent. In many cases, the Hankel singular value ( ( )) as well as eigenvalues of P, Q decay very rapidly [2]
Definition 2: (Moore 1981 Let G(s) be the asymptotically stable, minimal, system G(s) is called Lyapunov balanced if
∑ ( ) (4)
Where are the
multiplicities of and , are called
the Hankel singular values of G(s).
By balanced basis we obtain the states simultaneously has the property which are difficult to reach and difficult to observe. Hence, a simplified model is obtained by truncating the states which correspond to small Hankel singular values .
Theorem 2.1Let G(s) be asymptotically stable and minimal system so according to Lyapunov the G(s) have the balanced realization i.e.
( ) [ ] [ ]
With ∑ (∑ ∑ ) where,
∑ ( )
and ∑ ( )
Then the reduced-order model
( ) [
]
is asymptotically stable, minimal obtained by truncation and satisfies
‖ ( ) ( )‖ ( )
III. STOCHASTIC BALANCING METHOD
The asymptotically stable and minimal system G(s) given in (1) with
a) m = p, i.e., G(s) is square, and b) det(D) 0.
Let minimal phase left spectral factor of G(s) ( ) is given by W(s) i.e. ( ) ( ) ( ) where ( )
( ). A realization of W(s) can be computed as W(s)
=[ ] with , and
( ) where P is the controllability gramian of
G(s), and is the solution to the Riccati equation
( ) ( ) .
The G(s) have asymptotically stable, minimal, square and non-singular system is called stochastically balanced if
P = ( ), where
are the multiplicities of and
Hankel singular
value of so-called phase matrix
( ( )) ( )
Theorem 3.1 Let G(s) be the asymptotically stable, minimal, square and non-singular system is stochastically balanced and partitioned as in (2) with
( ) where
( )
and ( ). Then (s) is
asymptotically stable, minimal obtained by the stochastic balanced truncation and is satisfies
‖ ( )‖ ∏
‖ ( )‖ ∏
is asymptotically stable, minimal.
IV. BOUNDED REAL BALANCING
A bounded real system is one of the important classes of dynamical systems. The systems whose transfer function H satisfies ( ) ( ) which are stable is so-called bounded real system. G(s) in (1) is called bounded real if and ( ) ( )
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Definition4: It can be shown that G(s) is bounded real if and only if there exist a such that
(
)
(
)
( )
Where any solution of (4) lies between two extremal solutions, i.e. 0 is the unique solution to (4) such that
( ) is asymptotically stable. Define
. Then a dual Riccati equation
( ) ( )
( )
is obtained where As in the case for (4), any solution of (5) lies between two extremal solutions,
i.e. (4) and (5) are called the
bounded real Riccati equations of G(s). It is easy to show that if is a solution to (4), and then is a solution to (5).Hence and . Then a bounded real balancing transformation is obtained by balancing with , which is equivalent to balancing and
A bounded real system G(s) is called bounded real balanced if
( )
where , are
the multiplicities of , and . are called
the bounded real singular values of G(s).
Theorem 4.1 Let G(s) be bounded real balanced with the asymptotically stable, minimal, bounded real system and partitioned as in (2) with ( )
where
( )and ( .
Let the reduced order model (s) be obtained by truncation as in (2). Then (s) is asymptotically stable, minimal, bounded real balanced and satisfies
‖ ( ) ( )‖ ( )
V. POSITIVE REAL BALANCING METHOD
A positive real (passive) system is defined in a physical sense, i.e. positive realness is defined as the energy released by the system due to its internal process is always less than the energy librated by the system. Positive Real (PR) systems have been motivated by the study of linear electric circuits.
Definition 5:The asymptotically stable system
( ) * + ( ) ( )
is called positive real if
and ( ) ( )
where G(s) = ( ) Define
. In the sequel, we will examine only strictly
positive real systems, G(s) is positive real if and only if there exists a Ƒ = such that
( ) ( )) ( )
As in the bounded real case, a dual Riccati equation
A ( ) ( ) ( )
is obtained where (7) and (8) are the so-called positive real Riccati equation of G(s). Any solutions
and of, respectively, (7) and (8) lie between two extremal solutions, i.e. and
If is easy to show that if
is a solution to (7) then is a solution to (8). Hence and . Then analogously to bounded real balancing transformation is obtained by balancing the minimal solutions and to (7) and (8), respectively.
A positive real system is called balanced if
( ) where
are the multiplicities of the
positive real singular values of G(s).
The, denoted by M is defined as the Moebius transformation of a square bounded-real system H(s), is defined as
H(s) → G(s) =( ( )) ( ( )). (9)
It is well known that G(s) in (9) is positive real. M is a bijection with inverse
( )→ ( ) ( ( ) )( ( ) ) (10)
If G(s) is a positive real system, H(s) in (10) is a square bounded real system.
Theorem 5.1 Let the asymptotically stable, minimal, positive real system G(s) have the following positive real balanced realization with ( )
where ( ) and
( ).Then reduced order model
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Note that there exists no results on the norm of the error G(s) − (s). It is clear that the error results of the stochastic balancing can be employed for positive-real balancing as well.
Theorem 5.2For a given the asymptotically stable positive real system G(s), let (s) be obtained by the positive real balanced truncation i.e. Define ( ) ,
‖ ‖ ∑ ‖ ( )‖ Then,
‖( ( )) ( ( )) ‖ (11)
‖(( ( )) ) ( ( ) ( ))‖ (12)
We state (12) as a multiplicative-like error bound rather than an exact multiplicative error bound because of the term ( ). However, one can easily see that it is a multiplicative error in terms of ( ) and ( ).
VI. FREQUENCY WEIGHTED BALANCING METHOD
Among all the balancing methods discussed previous have main goal to approximate the full order model G(s)
over all frequencies. However, in several applications a main focus is concentrated over a certain frequency range. This problem leads to the so-called the frequency weighted balancing method.
Given the original full-order model
the input weighting
the output weighting matrix
,
our objective is to find a lower-order model ( ) such that‖ ( ) ‖
is made as small as possible. Assume that G, and have the following state-space realizations:
*
+, [ ] [ ]
With . Let us assume ( )⏟
→ subsequently otherwise it can be deleted by interchanging
Now the state-space realization for the weighted transfer matrix is given by
[
] [ ̅ ̅
̅ ̅]
Let ̅and ̅be the solutions to the following Lyapunov equations:
̅ ̅ ̅ ̅ ̅ ̅ ( )
̅ ̅ ̅ ̅ ̅ ̅ ( )
Then the input weighted Gramian P and the output weighted Gramian Q are defined by
[ ] ̅ * + [ ] ̅ * +
It can be shown easily that P and Q satisfy the following lower-order equations:
[ ] [
] [ ] [
] [ ] [ ]
[
] [ ] [ ] [
] [ ] [ ]
The computation can be further reduced if or
In the case of P can be obtained from
( )
While in the case of , Q can be obtained from
( )
Now let T be a nonsingular matrix such that
( ) [ ]
(i.e., balanced) with ( )
and ( ) and partition the
system accordingly as
*
+ [
]
Then a reduced-order model is obtained as
[ ]
In this case the approximation error is never be defined by the a priori error bound as in all the case previously done and there is no confirmation of stability of reduced-order model
Relative error model reduction problem is one o the frequent for the frequency-weighted model reduction problem where the objective is to find a reduced-order model so that
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And ‖ ‖ is made as small as possible. is usually called the relative error. This problem can be simply formulated for a system G which is square and invertible is given as,
⏟
‖ ( )‖
Of course, the dual approximation problem
( )
can be obtained by taking the transpose of G. A multiplicative approximation is also described by the approximation which is given below:
( )
Where is termed as multiplicative error. The relative and multiplicative approximations is obtained through the frequency-weighted balanced truncation method which produces the corresponding error bound
Theorem 6.1 Let G, be the asymptotically stable, minimal system G(s) having the square transfer matrix of nth-order with a state-space realization
*
+
Let P and Q be the solutions to
( )
( )
Suppose
( )
( )
With and let the realization of G be partitioned compatibly with , as
( ) [ ]
Then,
[ ]
is stable and minimum phase. Furthermore,
‖ ‖ ∏ ( (√ ) )
‖ ‖ ∏ ( (√ ) )
VII. CONCLUSIONS
In this paper we have considered a survey of balancing related model reduction schemes and their corresponding error norms.
We have introduced the different types of balancing method that can be used to characterize the importance of the balanced truncation model reduction method. A frequency weighted balanced reduction method with guaranteed stability and bound on the error system is presented.
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